An unnamed spokesperson for the Kurdistan Liberation Hawks, who have claimed to be behind a series of bombings in Turkey in recent years, said it carried out the attack on Thursday in the Bayrampasa district, not far from Istanbul airport.
The group is believed to have links to outlawed Kurdish rebels fighting security forces in southeast Turkey. It was not immediately possible to verify the authenticity of the report.
Celalettin Cerrah, an Istanbul police chief, confirmed that the explosion was caused by a bomb.
An official at Istanbul's Capa hospital told Reuters Zafer Isik, a 21-year-old civilian, had died from his wounds.
Bayrampasa's police told Reuters that 16 people were wounded, including a child. The official said none of them was in a serious condition.
Local people said the blast was very powerful.
Murat Seymen, a machine worker, told Reuters: "I heard a loud explosion and felt the ground move. It was like an earthquake. I ran to the shop and there was blood on the ground. One boy had lost a leg."
He said he helped carry the boy to a car to be taken to hospital, and another person lost an arm.
"I heard a loud explosion and felt the ground move. It was like an earthquake"
The blast occurred 100m from a local headquarters for riot police, who regularly used the cafe.
A dozen officers sealed off the side street where the cafe is located.
Militant groups including Kurdish separatists, Islamists and ultra-leftists have carried out attacks on civilian, security and military targets in Turkey in the past.
Turkey blames the Kurdistan Workers Party (PKK), which is viewed as a terrorist organisation by the European Union and the United States, for the deaths of more than 30,000 people since they started an armed struggle in 1984 for a Kurdish state.
Islamists set off a series of bombs in Istanbul in November 2003, attacking British and Jewish sites and killing more than 60 people.